The seamer was plagued by persistent back issues during the Ashes series and missed all three matches against the West Indies.
The 40-year-old had an operation this week on his anterior cruciate ligament and could be out for up to eight months.
The 30-year-old rejoined City in 2020 after a spell with Lyon, having previously played for them from 2014 to 2017.
Manchester United fans have been urged to grant Erik ten Hag patience by the club’s football director, John Murtough